CAPS 2008

Determine all triples $(x, y, z)$ of positive real numbers which satisfies the following system of equations $$2x^3=2y(x^2+1)-(z^2+1),$$ $$2y^4=3z(y^2+1)-2(x^2+1),$$ $$2z^5=4x(z^2+1)-3(y^2+1).$$

$ABCDEF$ is a convex hexagon, such that $|\angle FAB| = |\angle BCD| =|\angle DEF|$ and $|AB| =|BC|,$ $|CD| = |DE|$ , $|EF| = |FA|$ . Prove that the lines $AD$ , $BE$ and $CF$ are concurrent.

Find all primes $p$ such that the expression $$\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2$$ is divisible by $p^3$ .

Prove that there exists a positive integer $n$ , such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$ .

$ABCDE$ is a regular pentagon. Determine the smallest value of the expression $$\frac{|PA|+|PB|}{|PC|+|PD|+|PE|},$$ where $P$ is an arbitrary point lying in the plane of the pentagon $ABCDE$ .

Find all triplets $(k, m, n)$ of positive integers having the following property: Square with side length $m$ can be divided into several rectangles of size $1\times k$ and a square with side length $n$ .